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 1 

Proceedings of the 2007 ASME Rail Transportation Division Fall Technical Conference 

RTDF2007 

September 11-12, 2007 Chicago, Illinois, USA 

 RTDF2007-46009 

EQUATIONS OF MOTION FOR TRAIN DERAILMENT DYNAMICS 

D.Y. Jeong 

M.L. Lyons 

O. Orringer 

A.B. Perlman 

U.S. Department of Transportation 

Volpe National Transportation Systems Center 

Cambridge, Massachusetts, USA 

ABSTRACT 

This paper describes a planar or two-dimensional model to 

examine the gross motions of rail cars in a generalized train 

derailment. Three coupled, second-order differential equations 

are derived from Newton’s Laws to calculate rigid-body car 

motions with time. Car motions are defined with respect to a 

right-handed and fixed (i.e., non-rotating) reference frame. The 

rail cars are translating and rotating but not deforming. 

Moreover, the differential equations are considered as stiff, 

requiring relatively small time steps in the numerical solution, 

which is carried out using a FORTRAN computer code. 

Sensitivity studies are conducted using the purpose-built model 

to examine the relative effect of different factors on the 

derailment outcome. These factors include the number of cars 

in the train makeup, car mass, initial translational and rotational 

velocities, and coefficients of friction. Derailment outcomes 

include the number of derailed cars, maximum closing 

velocities (i.e., relative velocities between impacting cars), and 

peak coupler forces. Results from the purpose-built model are 

also compared to those from a model for derailment dynamics 

developed using commercial software for rigid-body dynamics 

called Automatic Dynamic Analysis of Mechanical Systems 

(ADAMS). Moreover, the purpose-built and the ADAMS 

models produce nearly identical results, which suggest that the 

dynamics are being calculated correctly in both models. 

NOMENCLATURE 

Dk = truck center spacing for the k

th rail car 

EC = lateral coupler force 

FC = coupler force 

FF = force at the front truck 

FR = force at the rear truck 

G = gap distance 

Ik = mass moment of inertia of the k

th rail car 

Lk = length between couplers for the k

th rail car 

LC = coupler length 

Mk = mass of the k

th rail car 

m = coupler moment 

n = number of rail cars in train makeup 

S = maximum swing angle for coupler 

u = velocity component in the y direction 

V = initial train speed 

v = velocity component in the x direction 

Wk = weight of the k

th rail car 

x = displacement in the x direction 

y = displacement in the y direction 

α = direction angle of truck 

β = direction angle of coupler 

γ = swing angle for coupler 

μ = coefficient of friction

θ = angular displacement of rail car 

Ω = initial angular velocity to initiate derailment 

INTRODUCTION 

Train derailments occur infrequently, but their 

consequences can be severe. Research is ongoing to develop 

methods to minimize and mitigate the damage resulting from 

train derailments. For example, accident modeling has been 

identified as a potential method to improve the crashworthiness 

of railroad tank cars [1]. Moreover, three recent accidents 

have focused attention on the structural integrity of railroad 

tank cars under impact loading conditions [2, 3, and 4]. 
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 2 

Sponsored by the Federal Railroad Administration (FRA), 

the Volpe National Transportation Systems Center is 

conducting and managing a research program to examine the 

structural integrity of railroad tank cars under accident loading 

conditions. The research program comprises three phases to 

assess the consequences of railroad tank cars involved in 

derailments. Each phase involves the development of 

computational models with different objectives: (1) dynamic 

modeling to estimate the gross motions of rail cars in a train 

derailment, (2) structural finite element modeling to calculate 

impact forces inferred from the gross motions, and (3) damage 

assessment modeling to estimate the deformations of the tank 

car before catastrophic failure occurs. 

In this paper, the differential equations of motion are 

derived for a generalized train derailment scenario. These 

differential equations describe planar (i.e., two-dimensional) 

rigid-body motions of the rail cars in a right-handed and fixed 

(i.e., non-rotating) reference frame. 

Previous work has been conducted to examine the gross 

motions of rail cars during a train derailment. For example, a 

special purpose model was developed in the 1970s [5] in which 

each car was assumed to behave as a rigid body with finite 

length but zero width. A major limitation of this model was 

that the cars were assumed to remain coupled during the 

derailment. 

Another special purpose model was later developed that 

allowed the cars to decouple, but the criterion for decoupling 

was incomplete [6, 7]. A planar model was later developed at 

Queen’s University [8], in which the differential equations of 

motion were derived from Lagrange’s equations and were 

solved numerically using a special-purpose computer program 

written in FORTRAN. 

Commercial general purpose software programs for multibody dynamics have also been used to examine the gross 

motions of rail cars in train derailments. For example, a threedimensional model of a 20-car train was developed using 

Dynamic Analysis Design Simulation (DADS) [9]. Another 

commercial program called Automatic Dynamic Analysis of 

Mechanical Systems (ADAMS) was used to develop a planar 

model for trains consisting up to 100 cars [10]. Results from 

the present work are compared directly to the ADAMS model. 

Explicit derivation and solution of the equations of motion can 

be used to provide a check on the motions calculated using the 

commercial programs, and vice versa. 

MATHEMATICAL FORMULATION 

Figure 1 shows the basic conventions for a right-handed, 

non-rotating description of the k

th rail car in a train consist, 

where k=1, 2...n, counting back (right to left in the figure) 

from the first car to derail. 

Figure 1. Rail Car Free-Body Diagram and Conventions 

The gross motion for each car is characterized by the 

coordinates (xk,yk,θk) at the center of mass. The car itself is 

defined by its length Lk between couplers and the distance Dk

between truck centers. The couplers are defined by a length LC. 

The coordinates and the car parameters may be used to 

determine the positions of the front and rear couplers: 

1

, 2 cos FCk k k k xxL = + θ 1, 2 sin FCk k k k yyL = + θ (1) 

1

, 2 cos RCk k k k xxL = − θ 1, 2 sin RCk k k k yyL = − θ (2) 

Figure 1 also shows the definitions and positive sign 

conventions for the forces acting on the k

th car. Ground forces 

FF k and FR k act at the front and rear trucks, respectively, at 

angles αF k and αR k with respect to the x-axis. The logic for 

determination of the ground forces and angles depends on 

whether the car has or has not derailed. That is, the model 

includes two types of friction: (1) on-track friction is an 

approximation of the maximum frictional force associated with 

emergency braking and (2) off-track friction is an estimate of 

the complicated resistive forces that are present when the car 

derails. 

If the car has derailed (xk equal to or greater than the X 

coordinate for the point of derailment), then FF k = FR k = 

μWk/2, where μ is the ground friction coefficient and Wk is the 

car’s rail weight. These are assumed to be Coulomb friction 

forces acting opposite to the instantaneous local velocities of 

the respective truck centers: 

FF k

FR k

FC k-1,k

FC k,k+1

EC k,k+1

EC k-1,k

mFC k

mRC k

xk

yk

θk

αF k

αR k

β D k-1,k k

Lk

LC

βk,k+1

Y

X
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 3 

( ) 1

tan Fk Fk F kα v u −= ( ) 1tan Rk Rk R kα v u −= (3) 

where 

1

2 sin F kk k k ux D = − θ θ   12 cos F kk k k vy D = + θ θ   (4) 

1

2 sin R kk k k ux D = + θ θ   12 cos R kk k k vy D = − θ θ   (5) 

are the truck center velocity components and the dotted 

quantities refer to derivatives with respect to time. 

Coupler forces are positive in buff and act at angles βk-1,k

and βk,k+1 with respect to the X-axis. The positions of adjacent 

car ends determine the angles: 

( )( ) 1

1, 1 1 tan / k k RC k FC k RC k FC k β yyxx −− −− =−− ⎡ ⎤ ⎣ ⎦ (6) 

( )( ) 1

,1 1 1 tan / k k RC k FC k RC k FC k β yy xx −+ ++ =− − ⎡ ⎤ ⎣ ⎦ (7) 

The same positions also determine the gaps between adjacent 

car ends: 

( )( ) 2 2

1, 1 1

1

2

G xx yy k k RC k FC k RC k FC k −− − = −+ − ⎡ ⎤ ⎣ ⎦ (8) 

( )( ) 2 2

,1 1 1

1

2

G xx yy k k RC k FC k RC k FC k ++ + = − +− ⎡ ⎤ ⎣ ⎦ (9) 

The equation of motion for the k

th car in the x-direction is: 

, 1 , 1 1, 1,

, 1 , 1 1, 1,

cos cos cos

sin cos

k k Ck k k k Ck k k k Rk Rk

Ck k k k Ck k k k

Mx F F F

E E

β β α

β β

+ +− −

+ +− −

=−−

+ −



 (10) 

The equation of motion for the k

th car in the y-direction is: 

, 1 , 1 1, 1,

, 1 , 1 1, 1,

sin sin sin

cos cos

k k Ck k k k Ck k k k Rk Rk

Ck k k k Ck k k k

My F F F

E E

β β α

β β

+ +− −

+ +− −

=−−

− +



 (11) 

The equation of motion for the typical k

th car in the θ-direction 

is: 

() ()

() ()

() ()

1

1

2 , 1 , 1 1, 1,

1

2 , 1 , 1 1, 1,

1

2 , 1

sin sin

cos cos

sin sin

k k RC k FC k

k Ck k k k k Ck k k k k

k Ck k k k k Ck k k k k

k Fk k k Fk Fk k Rk

I mm

LF F

LE E

DF F

θ

θβ θβ

θβ θβ

θα θα

+

+ +− −

+ +− −

+

= +

+ −+ −

+ −+ −

+ −+ −

⎡ ⎤ ⎣ ⎦

⎡ ⎤ ⎣ ⎦

⎡ ⎤ ⎣ ⎦



 (12) 

The coupler hardware between two cars comprises two sets 

of knuckle, shank, and draft gear. For the purposes of the 

present model, the two sets are treated as a single pin-ended 

bar, with the coupler force FC aligned along the bar, provided 

that neither individual swing angle limit has been reached. The 

bar may begin to support a moment at one end, when an angle 

limit is reached, and may support moments at both ends if both 

angle limits have been reached. Since the bar is also assumed 

to have no mass, force and moment equilibrium conditions 

applied to it require the presence of lateral forces, as well as the 

aligned force FC. Figure 2 shows the coupler free-body 

diagram, together with the reactions on the cars, for the typical 

coupler between car k and car k+1. 

Figure 2. Coupler Free-Body Diagram 

Figure 3 illustrates the coupler force curve assumed in the 

model. This characteristic includes a dead band or gap (0.25 

feet assumed) to simulate slack and free translation, which is 

centered about a nominal coupler length of 4 feet. The coupler 

stiffness is assumed to be 50×106

 lb/ft. 

θk+1

θk

βk, k+1

Gk, k+1

EC k, k+1

mFC k+1

mRC k

FC k, k+1

FC k, k+1

FC k, k+1

EC k, k+1

EC k, k+1

EC k, k+1
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 4 

Gap (ft)

3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5

Force (lb) -1.5e+7

-1.0e+7

-5.0e+6

0.0

5.0e+6

1.0e+7

1.5e+7

0.5 ft

0.25 ft

1.25 x 107 lb

Figure 3. Assumed Coupler Force Characteristic 

In addition to the aligned coupler forces ECk,k+1, Figure 1 

shows lateral forces EC k,k+1, a coupler moment mRC k at the rear 

end of car k, and a coupler moment mFC k+1 at the front end of 

car k+1. The positive sign conventions for the moments are 

defined such that they are in the directions shown when βk,k+1

exceeds θk and θk+1. For the front and rear couplers on the 

typical car k, the swing angles are: 

FCk k k k 1, γ β θ − = − RCk k k k , 1 γ β θ + = − (13) 

The moments are computed from these swing angles. 

The lateral force is derived from the moment equilibrium 

condition for the coupler: 

,1 1 ,1 ( )/ C k k RC k FC k k k E mm G + ++ = + (14) 

where Gk,k+1 is the current value of the coupler gap. 

Figure 4 shows the coupler moment characteristic assumed 

in the model. It includes a dead space between swings of 

±17.5o with a sudden rise when the swing angle is greater than 

the maximum swing. In practice, the maximum swing angle 

and moment would depend on the type of coupler (e.g., plainE, shelf-E, etc.). 

Initial conditions 

The train consist is initialized in a slack configuration. The 

point of derailment in the model is assumed to be the center of 

mass for the lead car. Therefore, with the first car center of 

mass at x1 equal to zero (i.e., point of derailment), the trailing 

car centers are set equal to 

( ) 1

1 0

1

1

for 2, 3

2

k

k j jj

j

x x LL G k n

−

+

=

=− + + + = ∑ " (15) 

where Go is equal to 4 feet. This equation describes the midslack condition, which is an arbitrary choice, but avoids the 

introduction of large initial coupler forces and has some degree 

of relation to real situations. The initial conditions for all car 

center coordinates (yk, θk) are zero. All initial gaps are 

Gk,k+1=G0. All other initial angles αF k, αR k, and βk,k+1 are zero. 

Swing Angle (degrees)

-20 -15 -10 -5 0 5 10 15 20

Moment (lb-ft) -1.0e+6

-8.0e+5

-6.0e+5

-4.0e+5

-2.0e+5

0.0

2.0e+5

4.0e+5

6.0e+5

8.0e+5

1.0e+6

Figure 4. Assumed Coupler Moment Characteristic 

Initial conditions for rate variables 

In general, the motion of the train in the generalized 

derailment presented here begins with an initial rotation 

imparted to the lead car and with all cars traveling at a given 

initial speed. Thus, the initial velocities are 

k x = V 0 k y = (16) 

for all cars where V is the initial train speed. The initial angular 

velocities are 

1 θ

 = Ω and 0 for 2, 3 k θ = = k n  " (17) 

where Ω is the initial rotational speed of the first car to start the 

pileup. 

With these initial conditions, the lead car swings out in the 

direction of the initial rotation, with higher off-track frictional 

forces applied at the trucks opposing the direction of motion. 

As the cars derail, the coupler forces between cars create a 

moment couple. The many highly coupled interactions cause 

the rail cars to form irregular buckling patterns. 

Numerical solution to the equations of motion is carried 

out using a Runge-Kutta algorithm with an adaptive time step 

[11], which is coded into a purpose-built FORTRAN computer 

program. 
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 5 

SENSITIVITY STUDIES AND RESULTS 

Sensitivity studies are conducted using the purpose-built 

model for derailment dynamics. Baseline values were assumed 

and varied one factor at a time to examine the relative effect of 

various factors on the derailment outcomes in terms of (1) the 

number of derailed cars, (2) the maximum closing speeds, and 

(3) peak coupler forces. Closing speed or velocity is the 

relative difference in speed or velocity between two impacting 

cars. Table 1 lists the baseline values assumed in the sensitivity 

studies. 

Table 1. Baseline Values for Sensitivity Studies 

Parameter Baseline Value 

Train length 60 cars 

Car weight 150,000 lb 

Initial translational velocity 37 miles per hour 

(mph) 

Initial angular velocity 0.2 radian per second 

On-track coefficient of friction 0.15 

Off-track coefficient of friction 0.50 

Each parameter was varied from its baseline value by ±20 

percent and ±50 percent while the remaining inputs, including 

car length and coupler characteristics, were set equal to their 

respective baseline values. Table 2 lists the range of values 

assumed for each factor in the sensitivity studies. 

Table 2. Range of Values for Sensitivity Studies 

Parameter Range 

Train length 30 to 90 cars 

Car weight 75 to 225 kips 

Initial translational velocity 18.5 to 55.5 mph 

Initial angular velocity 0.1 to 0.3 radian per sec 

On-track coefficient of friction 0.075 to 0.225 

Off-track coefficient of friction 0.25 to 0.75 

Figure 5 shows the relative effect of changing each 

parameter on the number of derailed cars calculated by the 

purpose-built model. The baseline case results in 13 cars 

derailing from the 60-car train. Clearly, train speed has the 

greatest effect on the number of derailed cars. Increasing the 

train speed by 50 percent over the baseline increases the 

number of derailed cars from 13 to 24. Decreasing the initial 

train speed to 50 percent below the baseline reduces the 

number of derailed cars from 13 to 6. Friction also has a 

significant effect on the number of derailed cars. The relative 

difference between ground and rail friction coefficients is 

proportional to the force to decelerate the train. Greater 

differences in friction coefficients lead to greater retarding 

forces which slow the train. Moreover, friction forces have an 

inverse relationship compared to the other inputs. That is, 

increasing friction reduces the derailment severity in terms of 

the number of derailed cars while increasing the value of the 

other parameters increases the number of derailed cars. 

Variation from Baseline (%)

-60 -40 -20 0 20 40 60

Normalized Number of Derailed Cars

(Variation/Baseline)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Number of Derailed Cars

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

Train Speed

Angular Velocity

Train Length

Car Mass

Ground Friction

Rail Friction

Figure 5. Effect of Factors on Number of Derailed Cars 

Figure 6 shows the relative effect of changing each 

parameter on the maximum closing velocity calculated using 

the purpose-built model. The baseline case results in a 

maximum closing velocity of 19 mph. Train speed has the 

most significant effect on maximum closing velocity. 

Increasing the initial train speed by 50 percent over the baseline 

increases the maximum closing velocity by almost 60 percent. 

Decreasing the initial train speed by 50 percent below the 

baseline reduces the maximum closing velocity by 30 percent. 

Ground friction has a somewhat moderate effect on closing 

velocity. Increasing ground friction by 50 percent over the 

baseline increases maximum closing velocity by more than 20 

percent. Changing the other input parameters varies the 

maximum closing velocity by at most ±15 percent. 

Variation from Baseline (%)

-60 -40 -20 0 20 40 60

Normalized Closing Speed (Variation/Baseline)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Maximum Closing Speed (mph)

0

5

10

15

20

25

30

35 Train Speed

Angular Velocity

Train Length

Car Mass

Ground Friction

Rail Friction
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 6 

Figure 6. Effect of Factors on Maximum Closing Velocity 

The relatively weak effect of initial rotational velocity on 

closing velocity means that the cause of the derailment (e.g., 

broken rail, wheel climb, excessive lateral-to-vertical wheel 

load ratio, etc.) does not have a significant effect on the 

maximum closing velocities in car-to-car impacts. 

If the coupler forces are relatively large, couplers can 

break and may potentially become impacting objects. Figure 7 

shows the relative effect of changing each parameter on the 

peak coupler force calculated by the purpose-built model. The 

peak coupler force corresponding to the baseline case is 1,821 

kips. Increasing the initial train speed by 50 percent over the 

baseline value increases the peak coupler force by almost 

twice. 

Variation from Baseline (%)

-60 -40 -20 0 20 40 60

Normalized Peak Coupler Force (Variation/Baseline)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Peak Coupler Force (kips)

0

1000

2000

3000

Train Speed

Angular Velocity

Train Length

Car Mass

Ground Friction

Rail Friction

Figure 7. Effect of Factors on Peak Coupler Force 

Repeating the sensitivity studies using the ADAMS model 

for derailment dynamics gives nearly identical results, which 

are presented in the appendix. 

CONCLUDING REMARKS 

This paper describes a purpose-built model to examine the 

dynamics of rail cars in a generalized train derailment. The 

equations of motion were derived explicitly from Newton’s 

laws of physics, and were coded into a built-for-purpose 

FORTRAN computer program. Sensitivity analyses were 

performed with the model to examine the relative effect of 

various factors on derailment severity in terms of the number of 

derailed cars, maximum closing speeds for colliding cars, and 

peak coupler forces that may potentially become impacting 

forces. 

The results from the purpose-built model were compared 

to those calculated using a commercial software code called 

ADAMS [10]. The computational times to run the purposebuilt FORTRAN model were much less than the ADAMS 

model (minutes versus hours). Moreover, the two models 

produced virtually identical results. This comparison provided 

verification of both analyses. 

The similarity of results from the purpose-built model 

presented in this paper and the ADAMS model suggests that 

the dynamics are being calculated correctly for the assumed 

scenarios in both models. However, the assumptions at best 

provide only a simplified representation of actual conditions. 

For example, it is unlikely that the actual terrain at a derailment 

site will be perfectly level. In addition, the effective friction 

between derailed cars and the ground is likely to vary. Thus, 

the models provide a general representation but do not 

reproduce specific derailment events. 

Nevertheless, the general results can provide some useful 

insights. For example, the closing velocities in post-derailment 

car-to-car impacts appear to average about half of the initial 

train speed. 

Another general result is that train speed appears to be the 

most important variable. The assumed train speed was 

increased by a factor of 3 (18.5 to 55.5 mph) in a series of 

cases, with the following outcomes: (1) the number of derailed 

cars increased by a factor of 4, (2) closing speed for potential 

collisions increased by a factor of 2.5, and (3) peak coupler 

force increased by a factor of 6. 

ACKNOWLEDGMENTS 

The Federal Railroad Administration (FRA), Office of 

Research and Development sponsored the work described in 

this paper. Ms. Claire Orth is the Chief of the Equipment and 

Operating Practices Division. Mr. Francisco Gonzalez is the 

project manager for research on railroad tank cars. Mr. Eloy 

Martinez also provides technical direction to this project. 

REFERENCES 

[1] Transportation Research Board, 1994: “Ensuring Tank Car 

Safety,” TRB Special Report 243, Washington, DC. 

[2] National Transportation Safety Board, 2004: “Derailment 

of Canadian Pacific Railway Freight Train 292-16 and 

Subsequent Release of Anhydrous Ammonia Near Minot, 

North Dakota January 18, 2002,” Railroad Accident Report 

NTSB/RAR-04/01. 

[3] National Transportation Safety Board, 2006: “Collision of 

Union Pacific Railroad Train MHOTU-23 with BNSF 

Railway Company Train MEAP-TUL-126-D With 

Subsequent Derailment and Hazardous Materials Release, 

Macdona, Texas, June 28, 2004,” Railroad Accident 

Report NTSB/RAR-06/03. 

 [4] National Transportation Safety Board, 2005: “Collision of 

Norfolk Southern Freight Train 192 with Standing Norfolk 

Southern Local Train P22 With Subsequent Hazardous 

Materials Release at Graniteville, South Carolina January 

6, 2005,” Railroad Accident Report NTSB/RAR-05/04. 
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for 

public release; distribution is unlimited. 

 7 

[5] Yang, T.H., Manos, W.B., Johnstone, B., 1972: “Dynamic 

Analysis of Train Derailments,” ASME Winter Annual 

Meeting, ASME Paper 72-WA/RT-5. 

[6] Birk, A.M., Anderson, R.J., Coppens, A.J., 1990: 

“Computer simulation of a derailment accident, Part I – 

Model basis,” Journal of Hazardous Materials 25, 121-

147. 

[7] Birk, A.M., Anderson, R.J., Coppens, A.J., 1990: 

“Computer simulation of a derailment accident, Part II – 

Sample simulation,” Journal of Hazardous Materials 25, 

149-165. 

[8] Toma, E.E., 1998: “A Computer Model of a Train 

Derailment,” Ph.D. Thesis, Queen’s University, Kingston, 

Ontario, Canada. 

[9] Han, H.S., Koo, J.S., 2003: “Simulation of train crash in 

three dimensions,” Vehicle System Dynamics 20, 435-450. 

[10] Paetsch, C.R., Perlman, A.B., Jeong, D.Y., 2006: 

“Dynamic Simulation of Train Derailments,” Proceedings 

of IMECE2006, 2006 ASME International Mechanical 

Engineering Congress and Exposition, IMECE2006-

14607. 

[11]Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, 

B.P., 1992: Numerical Recipes in FORTRAN: The Art of 

Scientific Computing, 2nd edition, Cambridge University 

Press, New York, NY. 

APPENDIX 

Figure 8 shows the relative effect of the different factors 

on the number of derailed cars. The purpose-built and 

ADAMS models calculate 13 derailed cars for the baseline 

case. The ADAMS model calculates slightly more derailed 

cars as the train speed increases than the purpose-built model 

(see Figure 5). The ADAMS results also show a slightly 

greater effect of friction on the number of derailed cars than the 

purpose-built model, which is attributed to different numerical 

methods used in the two models to solve the equations of 

motion. 

Variation from Baseline (%)

-60 -40 -20 0 20 40 60

Normalized Number of Derailed Cars

(Variation/Baseline)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Number of Derailed Cars

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30 Train Speed

Angular Velocity

Train Length

Car Mass

Ground Friction

Rail Friction

Figure 8. Effect of Factors on Number of Derailed Cars 

using ADAMS Model 

Figure 9 shows the variations in maximum closing speeds 

calculated using the ADAMS model, which are similar to those 

from the purpose-built model (see Figure 6). Both models 

calculate a maximum closing speed of 19 mph for the baseline 

case. 

Variation from Baseline (%)

-60 -40 -20 0 20 40 60

Normalized Closing Speed (Variation/Baseline)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Maximum Closing Speed (mph)

0

5

10

15

20

25

30

Train Speed 35

Angular Velocity

Train Length

Car Mass

Ground Friction

Rail Friction

Figure 9. Effect of Factors on Maximum Closing Velocity 

using ADAMS Model 

Figure 10 shows the relative effect of changing each factor 

on the peak coupler force calculated using the ADAMS model. 

For the baseline case, the ADAMS model calculates a peak 

coupler force of 1,858 kips (compared to 1,821 kips from the 

purpose-built model). Overall, the variations in peak coupler 

forces calculated by both models are similar (see Figure 7). 

Variation from Baseline (%)

-60 -40 -20 0 20 40 60

Normalized Peak Coupler Force (Variation/Baseline)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Peak Coupler Force (kips)

0

1000

2000

3000

Train Speed

Angular Velocity

Train Length

Car Mass

Ground Friction

Rail Friction

Figure 10. Effect of Factors on Peak Coupler Force using 

ADAMS Model 
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